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There are many situations, both in and outside of mathematics, where the process of doing and undoing helps you organize your activities and figure out how to reverse what you’ve done. In mathematics, it is often important to know how to undo an operation. Here are some examples from everyday life and mathematics:

**Note 2**

- School buses pick up children every morning and then drop them off in the same spots every afternoon. Routes are usually organized by a “first on, last off” routine.
- You put on socks and then shoes every morning, and you take off shoes and then socks every night.
- If you added 3 to a number and got 724, you can get your original number back by subtracting 3.

Sometimes you do things that can’t be undone:

- If the cover comes off the hot pepper shaker while you’re sprinkling it on the pizza, there’s not much you can do to undo the process.
- If you mix blue laundry detergent and water, you’d have a hard time separating them back into their original components.
- If you subtracted 10 from a number, then multiplied the result by itself, you wouldn’t be able to find, with certainty, the original number just from undoing the steps.

How can you tell that you wouldn’t be able to definitively find the original number in the numerical rule given above, in which 10 is subtracted from a number and then that number is multiplied by itself?

*Tip: See if you can find two different inputs whose outputs are the same. How would that make it impossible to find the original number?*

Give some examples from teaching, mathematics, or anywhere else where doing and undoing comes into play.

Give an example of something you wish you could undo, but the undoing is impossible.

**Note 2**

This exercise on “doing and undoing” will be our first look at functions.

Read through the examples of things that can be done and undone, and things that cannot be undone.

**Groups:** Discuss the examples as a large group. You may want to discuss the number puzzle that cannot be undone. Then, take five minutes to discuss Problems A2 and A3 in pairs.

**Problem A1**

The inputs 12 and 8 each lead to the output 4. If you only knew that the output was 4, it would be impossible to determine which of 12 and 8 was the correct input.

**Problem A2**

Some examples:

- Driving directions. Telling someone how to get somewhere usually allows them to figure out how to get back (if there are no one-way streets involved). In particular, to drive back you must take each road the opposite direction, and in reverse order (last road first).
- Packing and unpacking. Especially with commercially shipped packages, it can be difficult to re-pack a box without knowing where things were located before you unpacked it.
- In mathematics, many algebra problems involve “undoing” steps. For example, if 3x = 12, you can find x by undoing the multiplication step.

**Problem A3**

Lots of things can’t be undone easily, like throwing a water balloon, using gasoline in a car engine, or exploding fireworks.